# What is the value of $\int_0^\pi \frac{\sin[(n + \frac{1}{2})t]}{\sin(t)}dt$, where $n$ is a natural number?

What is the value of $\int_0^\pi \frac{\sin[(n + \frac{1}{2})t]}{\sin(t)}dt$, where $n$ is a natural number?

-
How are you getting it? – sourish Feb 14 '14 at 5:35
Is the denominator supposed to be $\sin(t/2)$? If so, see the wikipedia page on the Dirichlet Kernel. – Batman Feb 14 '14 at 5:46
The answer is not $\pi$. It is $\pi$ if the denominator is $\sin(t/2)$. – Mhenni Benghorbal Feb 14 '14 at 5:55
I agree with Mhenni. – Claude Leibovici Feb 14 '14 at 6:02
How to proceed? Anybody? – sourish Feb 14 '14 at 6:04

Notice that, the integrand has a pole of order one at $t=\pi$ as Laurent series shows that
$$-\sin \left( ( n+{1}/{2} ) \pi \right)\left( t- \pi \right)^{-1}-c_1 + c_2 \left( t-\pi \right) +O ( ( t-\pi ) ^{2} ).$$